An improved integration scheme for Mode-coupling-theory equations

TL;DR

This paper introduces a nonuniform discretization method for solving Mode-coupling theory equations, reducing computational cost while maintaining accuracy in modeling the glass transition.

Contribution

It proposes a novel nonuniform grid approach for MCT equations, improving computational efficiency over traditional equidistant grids.

Findings

Significant reduction in grid points needed for accurate solutions

Improved computational performance demonstrated on hard disk and sphere models

Accurate calculation of critical packing fractions and nonergodicity parameters

Abstract

Within the mode-coupling theory (MCT) of the glass transition, we reconsider the numerical schemes to evaluate the MCT functional. Here we propose nonuniform discretizations of the wave number, in contrast to the standard equidistant grid, in order to decrease the number of grid points without losing accuracy. We discuss in detail how the integration scheme on the new grids has to be modified from standard Riemann integration. We benchmark our approach by solving the MCT equations numerically for mono-disperse hard disks and hard spheres and by computing the critical packing fraction and the nonergodicity parameters. Our results show that significant improvements in performance can be obtained by employing a nonuniform grid.